JEE Advanced 2014 · Paper 2 · Mathematics Q60

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MathematicsList match+3 / -1Question 60 of 60

Let $f_1:\mathbb R\to\mathbb R$ , $f_2:[0,\infty)\to\mathbb R$ , $f_3:\mathbb R\to\mathbb R$ and $f_4:\mathbb R\to[0,\infty)$ be defined by
$f_1(x)=\begin{cases}|x| & \text{if } x<0,\\ e^x & \text{if } x\ge0;\end{cases}\qquad f_2(x)=x^2;$
$f_3(x)=\begin{cases}\sin x & \text{if } x<0,\\ x & \text{if } x\ge0\end{cases}$
and
$f_4(x)=\begin{cases}f_2(f_1(x)) & \text{if } x<0,\\ f_2(f_1(x))-1 & \text{if } x\ge0.\end{cases}$

List II

P
$f_4$ is
Q
$f_3$ is
R
$f_2\circ f_1$ is
S
$f_2$ is

List I

1
onto but not one-one
2
neither continuous nor one-one
3
differentiable but not one-one
4
continuous and one-one

P → 3; Q → 1; R → 4; S → 2

P → 1; Q → 3; R → 4; S → 2

P → 3; Q → 1; R → 2; S → 4

P → 1; Q → 3; R → 2; S → 4

Answer and worked solution

Official answer: D

The functions. $f_2\circ f_1(x)=x^2$ for $x<0$ and $e^{2x}$ for $x\ge0$ . So
$f_4(x)=\begin{cases}x^2&x<0\\e^{2x}-1&x\ge0.\end{cases}$
(The paper labels the left list "List II" and the right list "List I"; the code reads the same either way.)

P, $f_4$ : continuous at 0, range $[0,\infty)$ (onto its codomain). Not one-one: $x^2$ for $x<0$ repeats values of $e^{2x}-1$ . (Its one-sided derivatives at 0 are 0 and 2, so it is not differentiable.) Onto but not one-one → 1.
Q, $f_3$ : both pieces have derivative 1 at 0, so it is differentiable; $\sin x$ repeats values for $x<0$ . Differentiable but not one-one → 3.
R, $f_2\circ f_1$ : jumps from $0$ to $1$ at $x=0$ , so it is discontinuous; values above 1 are taken twice. Neither continuous nor one-one → 2.
S, $f_2=x^2$ on $[0,\infty)$ : continuous and one-one → 4.

(D) P → 1, Q → 3, R → 2, S → 4

Topics:Types of functions Continuity Differentiability

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JEE Advanced 2014 · Paper 2, Mathematics question 60